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Unformatted text preview: Stochastic Process 11/03/2006 Lecture 6 Fundamentals of Estimation II NCTUEE Summary In this lecture, I will discuss: • Interval Estimator • T Random Variable • Maximum Likelihood Estimation Notation We will use the following notation rules, unless otherwise noted, to represent symbols during this course. • Boldface upper case letter to represent MATRIX • Boldface lower case letter to represent vector • Superscript ( · ) T and ( · ) H to denote transpose and hermitian (conjugate transpose), respectively • Upper case italic letter to represent RANDOM VARIABLE 61 1 Confidence Interval (1) For an interval estimator [ L ( x ) ,U ( x )] of a parameter θ based on the observation x , we say that the confidence coefficient of this interval is 1 α if P h θ ∈ [ L ( x ) ,U ( x )] i ≥ 1 α, or we say [ L ( x ) ,U ( x )] is a (1 α ) × 100% confidence interval if P h L ( x ) ≤ θ ≤ U ( x ) i = 1 α. Note: The random quantity here is the interval (based on the obser vation x ), not the parameter θ . That is, the probability statements P £ L ( x ) ≤ θ ≤ U ( x ) / refers to x , not θ . Specifically, to find the proba bility, we actually need to find P h L ( x ) ≤ θ ≤ U ( x ) i = P h x : L ( x ) ≤ θ and θ ≤ U ( x ) i . (2) Let’s see how to specify a confidence interval of the mean μ for two cases. (a) Unknown mean, known variance Let X 1 , ··· ,X n be i.i.d. Gaussian variables with unknown mean μ and known variance σ 2 . The sample mean is a Gaussian random variable with ¯ X n ∼ N ( μ,σ 2 /n ) and ¯ X n μ σ/ √ n ∼ N (0 , 1) . We can specify an interval [ z,z ] within which the normalized sample mean has a probability P • z ≤ ¯ X n μ σ/ √ n ≤ z ‚ = Q ( z ) Q ( z ) = 1 2 Q ( z ) , where Q ( z ) = R ∞ z 1 √ 2 π e y 2 / 2 dy is the standard Qfunction. With simple algebraic efforts, the above can be rewritten as P • ¯ X n σz √ n ≤ μ ≤ ¯ X n + σz √ n ‚ = 1 2 Q ( z ) . (1) This means the interval [ ¯ X n σz √ n , ¯ X n + σz √ n ] contains μ with proba bility 1 2 Q ( z ). By letting α = 2 Q ( z ), we can find a corresponding z α/ 2 such that this interval is a (1 α ) × 100% confidence interval for μ . 62 (b) Unknown mean and unknown variance Let X 1 , ··· ,X n be i.i.d. Gaussian variables with unknown mean μ and unknown variance σ 2 . The confidence interval now becomes [ ¯ X n S n z √ n , ¯ X n + S n z √ n ] , where the variance σ 2 is replaced by the sample variance S 2 n . So, the probability of μ containing in this interval is P • ¯ X n S n z √ n ≤ μ ≤ ¯ X n + S n z √ n ‚ = P  z ≤ ¯ X n μ S n / √ n  {z } , T ≤ z ....
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 Winter '10
 GFung

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